论文标题
LAX逗号的有序集类别
Lax comma categories of ordered sets
论文作者
论文摘要
令$ \ mathsf {ord} $为（预）订购集的类别。与$ \ Mathsf {ord}/x $（其行为众所周知）不同，在文献中找不到有关Lax Comma 2-category $ \ MATHSF {ord} // x $的文献。在本文中，我们表明健忘的函数$ \ mathsf {ord} // x \ to \ mathsf {ord} $是拓扑时，并且仅当$ x $完成时。此外，在合适的假设下，$ \ mathsf {ord} // x $已完成，并且在$ x $为时，笛卡尔就关闭了。我们结束时分析了此类别中的下降。也就是说，当$ x $完整并且载笛子关闭时，我们表明，对于$ \ mathsf {ord} // x $中的形态，对于在$ \ mathsf {ord} $中的下降有效，在$ \ mathsf {ordsf} $中有效，对于$ \ mathsf {ord} $来说是有效的，对于$ \ math是必不可少的，是有效的。
Let $\mathsf{Ord} $ be the category of (pre)ordered sets. Unlike $\mathsf{Ord}/X$, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category $\mathsf{Ord} //X$. In this paper we show that the forgetful functor $\mathsf{Ord} //X\to \mathsf{Ord} $ is topological if and only if $X$ is complete. Moreover, under suitable hypothesis, $\mathsf{Ord} // X$ is complete and cartesian closed if and only if $X$ is. We end by analysing descent in this category. Namely, when $X$ is complete and cartesian closed, we show that, for a morphism in $\mathsf{Ord} //X$, being pointwise effective for descent in $\mathsf{Ord} $ is sufficient, while being effective for descent in $\mathsf{Ord} $ is necessary, to be effective for descent in $\mathsf{Ord} //X$.
